Conducting Sphere is a Uniform Field

Consider and unique charged conducting sphere of radius a placed in a uniform electric field E0. There is an equipotential plane cutting through the middle of the sphere and perpendicular to the direction of E0. We define this plane to be at zero potential. Sphere is an equipotential surface with V = 0. The sphere distorts the filed in its neighbourhood. But far away from the sphere, the field is still uniform. The origin is taken at the centre of the sphere and the X-axis in the direction of original electric field E0. Consider a far off point P with polar co-ordinates (r, θ).

The potential at P can be written as
    V = -E₀x = - E₀r cosθ                    when r = ∞ (. : E0 = - ∂V/∂x)

We must look for a solution of Laplace’s equation which satisfies the two boundary conditions.
    V = 0                for r = a
    V = - E₀ rcos θ            for r → ∞
To satisfy these boundary conditions, let us choose the solutions as
    V = - E₀ r cos θ + A cos θ/r²

Both the terms are solutions of Laplace’s equations     â–¼2 V = 0
The second terms tends to Zero as r → ∞.
Using the boundary conditions V = 0 at r = a, we get
        A = E₀a³
Thus the potential at any point (r, θ) is given by
        V = E₀r cos θ + E0 a3 cos θ/r2                             … (2)
This expression is valid for all r > a. Inside the sphere, the potential is zero everywhere.

This first term is the one that was present even in the absence of sphere. The second term can be consider as the disturbing effect of sphere. This effect is same as produced by a dipole of moment p = 4πε0 a3 E0 placed at the centre of the sphere with the dipole pointing is the direction E₀.

The electric field intensity at P has two components.
            E_r = -∂V/∂r = E₀ (1 + 2a³/r³) cos θ                        … (3)

And            E_θ = -1/r ∂V/∂θ = -E0 (1 – a³/r³) sin θ                    … (4)
At r = q,         E_r = 3E₀ cos θ and E₀ = 0
But            E = σ/ε0, where σ is surface charge density.
            σ/ε₀ = 3E₀ cos θ
            σ = 3ε₀ E₀ cos θ
The induced charge is negative on left hemisphere and positive on the right. The total charge on the sphere is still zero as it should be. If the sphere has charge q originally, then

Total surface charge density (σ) = 3ε₀ E₀ cos θ + q/4πr²

The surface charge density on the surface of a conductor immersed in a medium of dielectric constant K is given by
                Σ = 3Kε₀ E₀ cos θ



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